Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{3} \cdot 7 $
Sign $-0.990 + 0.135i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.63i·5-s + (1 − 2.44i)7-s + 1.50i·11-s − 5.91i·13-s − 5.14i·17-s − 5.24·19-s + 8.78i·23-s − 8.24·25-s − 8.91·29-s − 3.24·31-s + (8.91 + 3.63i)35-s + 37-s + 6.65i·41-s + 9.37i·43-s + 3.69·47-s + ⋯
L(s)  = 1  + 1.62i·5-s + (0.377 − 0.925i)7-s + 0.454i·11-s − 1.64i·13-s − 1.24i·17-s − 1.20·19-s + 1.83i·23-s − 1.64·25-s − 1.65·29-s − 0.582·31-s + (1.50 + 0.615i)35-s + 0.164·37-s + 1.03i·41-s + 1.43i·43-s + 0.538·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
\( \varepsilon \)  =  $-0.990 + 0.135i$
motivic weight  =  \(1\)
character  :  $\chi_{3024} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3024,\ (\ :1/2),\ -0.990 + 0.135i)\)
\(L(1)\)  \(\approx\)  \(0.2348839587\)
\(L(\frac12)\)  \(\approx\)  \(0.2348839587\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (-1 + 2.44i)T \)
good5 \( 1 - 3.63iT - 5T^{2} \)
11 \( 1 - 1.50iT - 11T^{2} \)
13 \( 1 + 5.91iT - 13T^{2} \)
17 \( 1 + 5.14iT - 17T^{2} \)
19 \( 1 + 5.24T + 19T^{2} \)
23 \( 1 - 8.78iT - 23T^{2} \)
29 \( 1 + 8.91T + 29T^{2} \)
31 \( 1 + 3.24T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 - 6.65iT - 41T^{2} \)
43 \( 1 - 9.37iT - 43T^{2} \)
47 \( 1 - 3.69T + 47T^{2} \)
53 \( 1 + 12.6T + 53T^{2} \)
59 \( 1 + 8.91T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + 3.63iT - 71T^{2} \)
73 \( 1 + 0.420iT - 73T^{2} \)
79 \( 1 - 4.47iT - 79T^{2} \)
83 \( 1 + 3.69T + 83T^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−9.355967346641528020401138503511, −7.941300995750498387068471255125, −7.56478170158659667901970074088, −7.07182680301077184346309167881, −6.15680653620846991378733793454, −5.38017777466585760447485269800, −4.33551911647025977865853874290, −3.37937468640842531682727621720, −2.83243655251238924370354528644, −1.60579689076358203675037801330, 0.06780184692659005415246426677, 1.68130484232095230832576882433, 2.14174526118513057126558490459, 3.87942344716980959721899398316, 4.40879217110672421917994815551, 5.20704723750588350709998978738, 5.98127914031540064141535701313, 6.62981958143922836842261090879, 7.919511994803870624343131725835, 8.504777698765510255922699706942

Graph of the $Z$-function along the critical line